The hypothesis of formation of comets as an accompaniment to formation of Uranus and Neptune from icy planetesimals is attractive for several reasons, but has suffered from long-standing problems regarding formation of the planets themselves. The history of this problem is reviewed, and recent results are described that may help solve it. Numerical simulations of planet growth show that when the system of planetesimals is no longer artificially constrained to a power-law size distribution, growth of planets may occur in reasonable time. An adeguate number of comet-sized bodies to populate the Oort cloud is not produced as collisional debris during the planet-building process. Rather, the comets are probably a remnant of the original planetesimal “building blocks” from which the planets grew.

The distribution of perihelion points of long-period comets is known to cluster towards the solar apex, and some authors ascribe it to north-south asymmetry in the distribution of observers. Validity or otherwise of this alleged selection effect is tested by randomly picking up the same number of perihelia in the southern (δ < 0) as those in the northern (δ > 0) hemisphere. It is shown that the observed clustering cannot be ascribed to the asymmetry of observers. Further, 67 comets which are new in Oort’s sense are tested similary. The character of their distribution is similar to that of all the known comets. It appears difficult to interpret the clustering in terms of a recent stellar disturbance of the Oort cloud.

Recent discoveries seem to indicate a catastrophic history of terrestrial evolution, explicable in terms of Oort cloud disturbance by molecular clouds in the Galactic disc. The problem of Oort cloud replenishment thus assumes considerable significance and reasons are given for supposing comet exchange takes place during actual penetration of molecular clouds. The number density of comets in molecular clouds, thereby implied, seems to suggest primary condensations of ≤103km in a dense precursor state of spiral arms. If chemical and/or isotopic signatures of comets should indicate an extra-Solar System source, the theory of terrestrial catastrophism may place new constraints on our understanding of the origin of molecular clouds.

The effects of encounters with massive nebulae on the long-period comet population are examined, paying particular attention to the uncertainties in the data. An earlier conclusion, that the long-period comet system is dynamically unstable, is upheld. Whether replenishment by unbinding from a dense inner comet cloud is a viable hypothesis awaits detailed modelling, but a qualitative discussion is given which argues tentatively against it. If comets occur in molecular clouds, however, their capture into temporarily bound Solar System orbits is a natural consequence of close encounters for realistic velocities and potentials. A large disturbance or capture may have occurred a few Myr ago as the Sun emerged from the Orion spiral arm.

The theory of a huge reservoir of comets (the “comet cloud”) extending to almost interstellar distances is analyzed, paying special attention to its dynamical stability, formation process and orbital properties of the incoming cloud comets. The perturbing influence of passing stars and giant molecular clouds is considered. Giant molecular clouds may be an important perturbing element of the comet cloud, although they do not seem to change drastically former studies including only stellar perturbations. The more tightly bound inner portions of the comet cloud, say within 104 AU, would have withstood the disrupting forces over the age of the solar system. The theory of a primordial comet origin in the outer planetary region close to Neptune’s orbit is specially analyzed. A primordial comet origin is consistent with the cosmogonic view that a large amount of residual material was ejected during the last stage in the formation of the Jovian planets. The smooth diffusion in the energy space of bodies scattered by Neptune guarantees that most of them will fall in the narrow range of energies close to zero (near-parabolic orbits) where passing stars and GMCs can act effectively on them. The long time scales of ~109 yr required for bodies scattered by Neptune to reach near-parabolic orbits would indicate that the buildup of the comet cloud was an event that took place long after the planets formed. Depending on the field of perturbing galactic objects, it is possible to conceive that most scattered comets were stored in rather tightly bound orbits (a ~l04 AU), favoring the concept of their dynamical survival over several billion yr. Alternative theories of comet cloud formation, e.g. in-situ origin or interstellar capture, are also discussed. The main difficulty of the in-situ theory is to explain how comets could accumulate at large heliocentric distances where the density of the nebular material was presumably very low. The interstellar capture theory also meets severe dynamical objections as, for instance, the lack of observed comets with original strongly hyperbolic orbits and the extremely low probability of capture under most plausible conditions. Since our knowledge of the structure of giant molecular clouds and their frequency of encounters with the solar system is still very uncertain, the concept of capture of transient comet clouds during such encounters can be advanced very little beyond the speculative stage. Some other dynamical properties of relevance to theories of origin and structure of the comet cloud are also reviewed. We mention, for instance, the distribution of perihelion points on the celestial sphere. There seems to be here a well established deviation from randomness, although the debate on whether or not there is a preference of the perihelion clustering for the vicinity of the apex of the solar motion is still unsettled. The alleged correlation with the solar apex may be biased by the preference of comet discoveries in the northern hemisphere. Deviations from randomness might be caused by very close stellar passages in the recent past. The excess of retrograde orbits among the observed “new” and young comets - mainly those with q ≳ 2 AU - is another well known dynamical feature. Such an excess may probably be accounted for by the combined action of planetary and stellar perturbations. Because of the decreasing action of planetary perturbations with increasing heliocentric distances, a significant increase in the rate of passages of long-period comets is predicted for the outer planetary region.

Empirical evidence about the size and the origin of the Oort’s cloud of comets is confronted with theories about its origin. The slow diffusion of the orbits of the “new” comets into the inner solar system implies a redefinition of the concept of “new” comet. A gradual transfer of orbital angular momentum occurs from the planets to the comets as the comets grow older on shorter period orbits. The observed retrograde to prograde ratio of the new comets is difficult to explain. Either it comes from a poorly understood observational bias, or from a neglected secular action of the Galaxy, or it implies a recent asymmetrical perturbation of the Oort’s cloud (less than 10−20 million years ago). The grazing incidence of a giant molecular cloud or an exceptionally close stellar passage would introduce such an asymmetry; this would also be true for the unseen hypothetical stellar companion of the Sun recently invoked to explain the periodicity of the geological extinction of species through violent cometary showers.

New studies of the dynamical evolution of cometary orbits in the Oort cloud are made using a revised version of Weissman’s (1982) Monte Carlo simulation model, which more accurately mimics the perturbation of comets by the giant planets. It is shown that perturbations by Saturn provide a substantial barrier to the diffusion of cometary perihelia into the inner solar system; Jupiter also. Perturbations by Uranus and Neptune are rarely great enough to remove comets from the Oort cloud, but do serve to scatter the comets in the cloud in 1/a. The new model gives a population of 1.8 to 2.1 × 1012 comets for the present-day Oort cloud, and a mass of 7 to 8 earth masses. Perturbation of the Oort cloud by giant molecular clouds in the galaxy is discussed, as is evidence for a massive “inner Oort cloud” internal to the observed one. The possibility of an unseen solar companion orbiting in the Oort cloud and causing periodic comet showers is shown to be dynamically plausible but unlikely based on the observed cratering rate on the earth and moon.

The stochastic processes involved in the evolution of a hypothetical cloud of comets are investigated. The cloud is assumed to be randomly perturbed by passing stars approaching the Sun at distance less than 1 pc. Within the frame of the impulse approximation we derive analytical expressions for the probability distribution of the impulse imparted to comets for both close and distant approaches. An application to the rate of ejection of comets is presented.

The possibility of an observational biasing in the distribution of cometary aphelia is confirmed by statistics. Furthermore, the tendency of aphelia to form clusters and their distribution with respect to different coordinate systems is investigated.

The physical processes which may affect the evolution of meteor streams are discussed and a review is then given of the work carried out to date on the evolution of meteor streams. It is clear that they evolve principally due to the effect of planetary perturbations and radiation pressure. The formation of streams from the breakup of a comet is also discussed. All the evidence, including the recent, discovery of 1983TB points to the correctness of this hypothesis.

It has long been realised that Jovian perturbation is the dominant cause of the transition of long period comets (Period > 200 yr) into short period ones (P < 200 yr). When the differences in the detectability of comets in the two groups are taken into account it is clear that the present day flux of long period comets is sufficient to provide the present collection of short period comets in the inner solar system.

The fact that meteoroid streams are produced by decaying short period comets was first recognised around 1866 (see Hughes 1982a). The magnificent display of Leonids in that year enabled the radiant position and time of maximum rate to be easily calculated. Assuming the orbital period to be 33.25 yr Le Verrier (1867) and Schiaparelli (1867) published orbits for the meteoroid stream. The orbit of comet 1866 I, which had been discovered by Guillaume Tempel, from Marseilles on December 19, 1865 and independently by Horace P. Tuttle from Harvard, Massachusetts on January 5, 1866, has been calculated and published by Oppolzer (1867a). Almost to a man Peters (1867), Schiaparelli (1867) and Oppolzer (1867b) realised that the comet and the stream had similar orbits. Since that time many more examples have been put forward, two famous ones being the Perseids and comet Swift-Tuttle (1862 III) and the Eta Aquarids and Orionids both of which have comet Halley (1910 II) as their parent. For more details see Cook (1973).

It is now well known that object 1983 TB, discovered by IRAS, has an orbit very similar to that of the Geminid meteor stream. Calculations show that this orbit crossed over the orbit of Venus about 500 years ago. We will describe calculations tracing the history of both the object and the stream through this interaction with Venus and the present interaction with the Earth.

Various aspects of comet/asteroid distinctions and interrelations are reviewed with emphasis on recent work and paying special attention to the following problems: characteristics of cometary activity at large heliocentric distance and uniqueness of comet P/Schwassmann-Wachmann 1 with respect to physical properties, the rôle of Trojans and other small bodies in the outer planetary system concerning comet/asteroid classification, possibilities for physical evolution of comets into asteroids, orbital and dynamical overlap of the comet and asteroid populations, and the cometary versus asteroidal origin of Earth-approaching asteroids. With regard to these latter questions it is argued that recent discoveries indicate a more substantial probability for Jupiter family comets to develop into asteroidal objects than earlier believed, and several examples of cometary association for newly discovered Apollo-Amor asteroids are also referred to. However, the fractional cometary contribution to the traditional Apollo-Amor asteroid population (aphelia far inside Jupiter’s orbit) apparently can not yet be reliably estimated.

The spin rate distributions of comet nuclei and small asteroids are compared, and it is shown that Whipple’s (1982) finding of a faster average rotation for the asteroid sample was due to observational biases. In fact, the presently available rotational data do not exhibit any clear differentiation among comet nuclei and asteroids, except possibly for a higher abundance of short rotational periods among the Apollo-Amor objects.

We investigated the orbital evolution of Quadrantid-like meteor streams situated in the vicinity of the 2/1 resonance with Jupiter. For the starting orbital elements we took the values of the orbital elements of the Quadrantid meteor stream except for the semi-major axis which was varied between a = 3.22 and a = 3.34 AU. We considered these meteor streams as a ring and we investigated the resonant effect on the dispersion of this ring over a period of 13 000 years. Only gravitational forces due to the Sun and due to Jupiter were taken into account.

Saturn’s ring particles and airless moons are exposed to a large flux of interplanetary debris, principally comets and comet dust. Collisions with this debris are responsible for both physical and dynamical changes in objects orbiting about Saturn. Physical changes include cratering of large bodies and catastrophic disruption of small bodies. Dynamical changes, which are analyzed in this paper, include orbital alteration (principally of ring particles) and changes in spin state (which are only important for moons, as ring particle spins are continually altered by mutual collisions).

Saturn’s rings are rapidly being eroded by impacts of hypervelocity meteoroids in cometary orbits. Ejecta from these impacts will, in most cases, remain in orbit about Saturn and eventually be reaccreted by the rings, possibly at a different radial location. The resulting mass transport has been suggested as the cause of some of the features observed in Saturn’s rings (see Durisen 1984 for a review). Previous attempts to model this transport have used numerical simulations which have not included effects of angular momentum transport coincident with this mass transport. I have developed an analytic model for ballistic mass transport in Saturn’s rings. The model includes the effects of angular momentum advection and shows that the net material movement due to the combined effects of mass and angular momentum transport is roughly half that calculated when angular momentum advection is ignored. See Lissauer (1984) for further details.

All of Saturn’s mid-sized moons are rotating synchronously with their orbital period; thus, the same hemisphere of these moons always faces the planet, and the same point is always at the center of the satellite’s leading hemisphere (the apex). The satellites orbit Saturn with velocities ranging from 14 km/sec for Mimas to 8.5 km/sec for Rhea and 3.3 km/sec for Iapetus. These speeds are a significant fraction of the encounter velocities between comets and the Saturn system (~ 10−25 km/sec); thus, due to a type of “windshield effect” (more raindrops hit the windshield of a moving car than hit the rear window), more comets will collide with the moons’ leading hemispheres than with their trailing hemispheres; also, higher relative velocities between comets and the moons will lead to larger craters for impacts by comets of a given mass on the leading hemispheres. The combination of these effects suggests that regions of the satellites’ surfaces near the apex should be much more heavily cratered than regions near the antapex (Shoemaker and Wolfe 1982). Such a major cratering asymmetry has not been observed (Plescia and Boyce 1983). A similar situation exists for Jupiter’s moons Ganymede and Callisto (Passey and Shoemaker 1982).

McKinnon (1981) suggested that stochastic reorientation of the moons by impact-induced “spinup” during the establishment of the cratering record tended to equilibrate the crater densities between hemispheres. I have re-examined the dynamics of this problem, and I conclude that most impacts large enough to have caused “spinup” would have catastrophically disrupted the moons in question (Lissauer 1985).

This describes our integrator RADAU, which has been used by several groups in the U.S.A., in Italy, and in the U.S.S.R. over the past 10 years in the numerical integration of orbits and other problems involving numerical solution of systems of ordinary differential equations. First- and second-order equations are solved directly, including the general second-order case. A self-starting integrator, RADAU proceeds by sequences within which the substeps are taken at Gauss-Radau spacings. This allows rather high orders of accuracy with relatively few function evaluations. After the first sequence the information from previous sequences is used to improve the accuracy. The integrator itself chooses the next sequence size. When a 64-bit double word is available in double precision, a 15th-order version is often appropriate, and the FORTRAN code for this case is included here. RADAU is at least comparable with the best of other integrators in speed and accuracy, and it is often superior, particularly at high accuracies.

The Long-Term Evolution Project (LTEP), realized in collaboration by the IAS-Reparto di Planetologia (Rome, Italy) and the Astronomical Institute of SAV (Bratislava, Czechoslovakia), has been developed with the aim of giving a general insight into the dynamical evolution of short-period comets. The motion of all the known short-period comets has been investigated over a long time span (over 800 years) taking care, as far as possible, to eliminate the sources of possible discrepancies within the computations. An internally consistent data-set and an atlas of orbital evolutions are the first outputs of this project. The main characteristics of the LTEP are discussed, together with some general remarks on its importance for cometary studies, its limitations and the future developments.

The methods to estimate the integration errors, including the effects of truncation, rounding off, and instability of the solutions, are discussed. Polynomial error accumulation depends upon numerical method, stepsize, orbital period and also eccentricity; it is also machine dependent. Comets correspond to the most difficult case of exponentially diverging orbits; however they can be very close to resonant ordered regions.